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Theorem dsmm0cl 18947
Description: The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmcl.p  |-  P  =  ( S X_s R )
dsmmcl.h  |-  H  =  ( Base `  ( S  (+)m  R ) )
dsmmcl.i  |-  ( ph  ->  I  e.  W )
dsmmcl.s  |-  ( ph  ->  S  e.  V )
dsmmcl.r  |-  ( ph  ->  R : I --> Mnd )
dsmm0cl.z  |-  .0.  =  ( 0g `  P )
Assertion
Ref Expression
dsmm0cl  |-  ( ph  ->  .0.  e.  H )

Proof of Theorem dsmm0cl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dsmmcl.p . . . 4  |-  P  =  ( S X_s R )
2 dsmmcl.i . . . 4  |-  ( ph  ->  I  e.  W )
3 dsmmcl.s . . . 4  |-  ( ph  ->  S  e.  V )
4 dsmmcl.r . . . 4  |-  ( ph  ->  R : I --> Mnd )
51, 2, 3, 4prdsmndd 16155 . . 3  |-  ( ph  ->  P  e.  Mnd )
6 eqid 2454 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
7 dsmm0cl.z . . . 4  |-  .0.  =  ( 0g `  P )
86, 7mndidcl 16140 . . 3  |-  ( P  e.  Mnd  ->  .0.  e.  ( Base `  P
) )
95, 8syl 16 . 2  |-  ( ph  ->  .0.  e.  ( Base `  P ) )
101, 2, 3, 4prds0g 16156 . . . . . . . . . 10  |-  ( ph  ->  ( 0g  o.  R
)  =  ( 0g
`  P ) )
1110, 7syl6eqr 2513 . . . . . . . . 9  |-  ( ph  ->  ( 0g  o.  R
)  =  .0.  )
1211adantr 463 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g  o.  R )  =  .0.  )
1312fveq1d 5850 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  (  .0.  `  a
) )
14 ffn 5713 . . . . . . . . 9  |-  ( R : I --> Mnd  ->  R  Fn  I )
154, 14syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
16 fvco2 5923 . . . . . . . 8  |-  ( ( R  Fn  I  /\  a  e.  I )  ->  ( ( 0g  o.  R ) `  a
)  =  ( 0g
`  ( R `  a ) ) )
1715, 16sylan 469 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  ( 0g `  ( R `  a ) ) )
1813, 17eqtr3d 2497 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  (  .0.  `  a )  =  ( 0g `  ( R `  a )
) )
19 nne 2655 . . . . . 6  |-  ( -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  (  .0.  `  a )  =  ( 0g `  ( R `
 a ) ) )
2018, 19sylibr 212 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  -.  (  .0.  `  a )  =/=  ( 0g `  ( R `  a )
) )
2120ralrimiva 2868 . . . 4  |-  ( ph  ->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
22 rabeq0 3806 . . . 4  |-  ( { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/)  <->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
2321, 22sylibr 212 . . 3  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/) )
24 0fin 7740 . . 3  |-  (/)  e.  Fin
2523, 24syl6eqel 2550 . 2  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin )
26 eqid 2454 . . 3  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
27 dsmmcl.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
281, 26, 6, 27, 2, 15dsmmelbas 18946 . 2  |-  ( ph  ->  (  .0.  e.  H  <->  (  .0.  e.  ( Base `  P )  /\  {
a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
299, 25, 28mpbir2and 920 1  |-  ( ph  ->  .0.  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   (/)c0 3783    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   Basecbs 14719   0gc0g 14932   X_scprds 14938   Mndcmnd 16121    (+)m cdsmm 18938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-hom 14811  df-cco 14812  df-0g 14934  df-prds 14940  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-dsmm 18939
This theorem is referenced by:  dsmmsubg  18950
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